By Sebastian Hayes

Desert Island Numbers : The Number Ball, Number Marks and Number Bearers

November 15, 2012

(New readers may find it useful to read the preceding post first.)

The ‘Number Ball’

For my island paradise awaiting its Robinson Crusoe or Raffles I hit upon the idea of a clay ‘Number Ball’. The advantage of this device is that, apart from being portable, it allows one to get rid of a number once it is of no further interest and start again. A native might be sent, for example, by a chief to find out how many palm trees there were on a particular beach. Equipped with his Number Ball issued at the Central Data Hut he would arrive at the site and tear off as many little bits of clay as there were trees. He would report back to Central Office where the bits of clay would be recorded by an equivalent amount of scratches on a bone or knots in a cord, and he would then squash everything together to recover the original ball.
This system has an interesting feature : it is two-way in the sense that you can use the same apparatus for recording data but can then ‘de-record’ (wipe out) the data to recover the original set-up and start again. This means, firstly, that there is no wastage. There is also something aesthetically satisfying about such a simple apparatus having an ‘inverse’ procedure built into it : once you have completed your task, the Number Ball is returned to what it was in the beginning like the visible universe being absorbed back into the Tao from which it sprang.
Most recording systems do not have this feature : if you make a scratch on a bone you cannot ‘de-record’ without damaging the recording device, and crossing out something written with pen and ink is both messy and inefficient (in films a crossed out line often gets deciphered and leads to the conviction of a criminal). Destroying data has in fact become a considerable problem in modern society, hence the sale of shredders and civil servants’ perpetual fear of e-mails being picked up.
Clay Number Balls would be too messy for modern interior use but Blu-Tack is an alternative I have experimented with a little. There is, however, a certain risk of the little bits of clay or Blu-Tack sticking together and thus falsifying the reckoning.
The Number Ball is something of an anomaly mathematically and even philosophically. The object-numbers produced, i.e. the little bits of clay, do not strictly fulfil the requirement that number objects should not merge on being brought into close proximity — they can be made to merge or kept apart at will, so we have an interesting intermediate case somewhat comparable to that of semi-conductors.
Also, and this is more significant, the Number Ball is not, properly speaking, a number but rather a source of numbers, a number generator. In this respect it resembles an algebraic formula since the latter is not in itself a number (in any sense) but can be made to spew out numbers, as many numbers as you require. (For example the formula f(n) = (2n –1) gives you the odd numbers (counting 1) if you turn the handle by fitting in 1, 2, 3….. for n e.g. (2 × 1) – 1 = 1; (2 × 2) – 1 = 3; (2 ×3) – 1 = 5 and so on.)
Yet a Number Ball is not a formula or an idea : it remains an object. Of course, one could also call a box of matches or a set of draughtsmen ‘number generators’ but there is a difference here : the object-numbers are present in the box as distinct items (matches, counters) and are thus already numbers at least potentially, whilst bits of clay of Blu-Tack are not. A Clay Number Ball is actually a special type of generator since everything it produces comes from within itself and can be returned to itself. I have coined a term for this particular case : I call such an object an Aullunn. Although there are no complete Aullunn Generators in nature — not even, seemingly, the universe itself — many natural phenomena approximate to this condition. The varied life in and around a pond to all intents and purposes emerges spontaneously ‘from inside’ and dies back into it each winter; though we know that without some interaction with the environment, especially with sunlight, no generation would be possible.

Surprisingly I have not come across any accounts of tribes using clay Number Balls.

Number Marks and Number Bearers

A very different method of producing a set of numbers is to have an object or substance which is not itself a number (or a number generator) but a ‘bearer of numbers’ : the numbers are marks on the surface of the number bearer or deformations of it. This system, which at first sight seems a lot closer to the written system we use today, is extremely ancient and possibly pre-dates the widespread use of distinct number objects. The markings on the Ishango Bone, which dates back to about 20 000 B.C., are thought by archaeologists to have numerical significance. Other bones have been found dating almost as far back with scratches on them that are thought by some to indicate the number of kills to a hunter’s credit — one thinks at once of Billy the Kid, the “boy who had so many notches on his gun” (or was it Davy Crockett?).
The limitation of the notch system is that an incision is permanent which means that once the ‘number-bearer’ gets filled up it has to be stored somewhere or discarded like a diary. It thus tends to be used in rather special circumstances, either when one does not expect to be dealing with large quantities (rivals killed) or when one wants the information recorded to be permanent as, for example, in the case of inscriptions on State monuments.
Making charcoal marks on a wall, also an ancient practice, is ‘two-way’ in that one can rub out what one has written but the system would not be reliable for long-term recording of data because of effects of weather, flaking of surface &c. But numbers on a number bearer do not have to be marks : they can be reversible deformations, the prime example being knots in a cord. The great advantage of such systems is that, though very long lasting if the material is itself durable, the numerical data can easily be got rid of when no longer needed since knots can be untied. On the other hand because they take a lot of time to tie and untie, knots are unsuitable for rapid calculation and it would seem that the Inca State officials used quipus for storing data whilst they had some form of a counting-board system for calculations. Knots in a cord constitute a partial ‘two-way’ recording system — what is done can be undone — but they are at the same time quasi-permanent, indeed are in a sense the arithmetical equivalent of semi-conductors.

Knotted cords were in widespread use all over the world at one time and it is thought that mankind may even have gone though a ‘knotted cord’ era. Lao-Tse, the author of the Tao Te Ching (VIth century B.C.) who was a Luddite hostile to new-fangled inventions and to civilization generally speaks nostalgically of the days when mankind used knotted cords instead of written numbers.
In practice both systems are required, a ‘two-way’ number system which allows one to carry out calculations and then to efface them, plus a more permanent system which is used to record results if they are considered important enough. Thus the Incas (so it is thought) used quipus for permanent or semi-permanent records while they used stones and a counting board for calculation. The lack of a suitable ‘number-bearer’ to receive marks meant that inscribed number systems were a rarity until comparatively recently — baked clay tablets and papyrus were reserved for the bureaucratic elite and paper, a Chinese invention, only entered Europe in the latter Middle Ages and was expensive. Traders, even money-lenders and bankers, when they did not use finger-reckoning of which more anon, used a two-way system, namely counters and counting boards, right into the Renaissance. The abacus, a two-way system, was never widespread in Europe for some reason except in Russia, but in the East has remained in use right through to modern times. The soroban or Japanese abacus is still used today and as late as the nineteen-fifties a Japanese clerk armed with a soroban competed successfully with an American naval rating using an early electronic calculator. However, it must be pointed out that the Japanese achievement with the soroban depends on extensive practice in mental arithmetic rather than any particular merits of the device itself.
The drawback of a ‘two-way’ system such as an abacus where you erase as you go is that you cannot check for mistakes and even the result itself, once reached, has to be erased when we perform our next calculation i.e. there is no inbuilt recording element, no memory. But when there is no easy way of erasing we oscillate wildly between conservation and destruction : we tend to accumulate a vast amount of stuff, then periodically have a sort out and throw it all away, the pearls with the dross. Like most authors and mathematicians from time to time I have to tip out the entire contents of a large dustbin to search for a scrap of paper with some idea or formula written on it.
The principal drawback of a one-way semi-permanent system such as ink on paper is that it is incredibly wasteful and was until recently so expensive that the bulk of the world’s population, the peasantry, practically never used it and employed a pocket knife and a flat piece of wood to record data. Even in the computer era we still use the chalk and blackboard two way system though the chalked notice-board in the hall of buildings or private residences — to mark who is in or out — which was once commonplace is now virtually a thing of the past. I myself buy rolls of lining paper (which I clip down over a table) partly because I like to have plenty of room for drawing and calculation but also partly for reasons of economy — you get a lot of paper in a roll compared to an exercise book. It is a sobering thought that no less than a hundred years ago Ramanujan, one of the greatest names in Number Theory, like so many other Indian mathematicians of the time worked with slate and chalk because he found paper too expensive. Although to my knowledge no one has suggested this, I would guess that this is one of the main reasons why his early mathematical writings are so hard to follow — he left no tracks because he generally just copied out his conclusions, then literally wiped the slate clean (Note 1). To many people the results seemed to come from nowhere and indeed he was often incapable of explaining how he got them. (Ramanujan lived a century too early : today we have an improved ‘chalk and board’ system, the Whiteboard. At last marks can be easily erased without mess. I use large boards everyday and have somewhat moved on from lining paper to a more up to date recording system.)

To be continued

Note 1 A brief article on Ramanujan “Is there a Ramanujan problem?” reprinted from an edition of the magazine M500 can be found on my website www.sebastianhayes.co.uk